Improved Resource-bounded Borel-cantelli and Stochasticity Theorems

This note strengthens and simpliies Lutz's resource-bounded version of the Borel-Cantelli lemma for density systems and martingales. We observe that the technique can be used to construct martingales that are \additively honest," and also martingales that are \multiplicatively honest." We use this to improve the \Weak Stochasticity Theorem" of Lutz and Mayordomo: their result does not address the issue of how rapidly the bias away from 1/2 converges toward zero in a \stochastic" language, while we show that the bias must vanish exponentially.